.
(a) λ1=4,λ2=−1,x1=(3,2)T;
(b) λ1=8,λ2=3,x1=(1,2)T;
(c) λ1=7,λ2=2,λ3=0,x1=(1,1,1)T
.
(a) λ1=3,λ2=−1,x1=(3,1)T;
(b)
λ1=2=2 exp(0),λ2=−2=2 exp(π,i),x1=(1,1)T;
(c)
λ1=2=2 exp(0),λ2=−1+3–√i=2 exp(2πi3),λ3=−1−3–√i=2 exp(4πi3),x1=(4,2,1)T
. x1=70,000,x2=56,000,x3=44,000
. x1=x2=x3
. (I−A)−1=I+A+⋯+Am−1
.
(a) (I−A)−1=⎡⎣⎢100−10−1312⎤⎦⎥;
(b)
A2=⎡⎣⎢000−200200⎤⎦⎥,A3=⎡⎣⎢000000000⎤⎦⎥
. (b) and (c) are reducible.
. Hint: See Theorem6.8.2 .
.
(b) Hint: Apply Perron’s theorem to B and C.
11. Hint: Make use of the result from Exercise 10.
12. Hint: Apply Perron’s theorem to Ak.
13.
(c) Hint: Show that if c1 did equal 0, then yj would approach the zero vector as j→∞, contradicting the result from part (b).
15.
(d) w=≈(1229,1229,329,229)T(0.4138,0.4138,0.4138,0.1034,0.0690)T
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